Number Theory pp 179-192

# The Elementary Proof of the Prime Number Theorem: An Historical Perspective

• D. Goldfeld
Chapter

## Abstract

The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x > 1, let π(x) denote the number of primes less than x. The prime number theorem is the assertion that
$$\mathop {\lim }\limits_{x \to \infty } \pi \left( x \right){\raise0.7ex\hbox{{}} \!\mathord{\left/ {\vphantom {{} {}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{{}}}\frac{x} {{\log (x)}} = 1.$$
This theorem was conjectured independently by Legendre and Gauss.

## Keywords

Prime Number Elementary Proof Tauberian Theorem Joint Paper Prime Number Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [A-B-G]
K.E. Aubert, E. Bombieri, D. Goldfeld, Number theory, trace formulas and discrete groups, symposium in honor of Atle Selberg, Academic Press Inc. Boston (1989).
2. [A-S]
N. Alon, J. Spencer, The probabilistic method, John Wiley & Sons Inc., New York (1992).
3. [B]
H. Bohr, Address of Professor Harold Bohr, Proc. Internat. Congr. Math. (Cambridge, 1950) vol 1, Amer. Math. Soc, Providence, R.I., 1952, 127–134.Google Scholar
4. [Ch]
J. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176.
5. [C-G]
F. Chung, R. Graham, Erdos on graphs: his legacy of unsolved problems, A.K. Peters, Ltd., Wellesley, Massachusetts (1998).Google Scholar
6. [C]
L.W. Cohen, The annual meeting of the society, Bull. Amer. Math. Soc 58 (1952), 159–160.Google Scholar
7. [D]
H.G. Diamond, Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. vol. 7 number 3 (1982), 553–589.
8. [E]
P. Erdos, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, Proc. Nat. Acad. Scis. U.S.A. 35 (1949), 374–384.
9. [G]
D. Goldfeld, The Erdos-Selberg dispute: file of letters and documents, to appear.Google Scholar
10. [H1]
J. Hadamard, Étude sur les proprietés des fonctions entiéres et en particulier d’une fonction considérée par Riemann, J. de Math. Pures Appl. (4) 9 (1893), 171–215; reprinted in Oeuvres de Jacques Hadamard, C.N.R.S., Paris, 1968, vol 1, 103–147.Google Scholar
11. [H2]
J. Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses consequences arithmétiques, Bull. Soc. Math. Prance 24 (1896), 199–220; reprinted in Oeuvres de Jacques Hadamard, C.N.R.S., Paris, 1968, vol 1, 189–210.
12. [Ho]
P. Hoffman, The man who loves only numbers, The Atlantic, November (1987).Google Scholar
13. [I]
A.E. Ingham, Review of the two papers: An elementary proof of the prime-number theorem, by A. Selberg and On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, by P. Erdös. Reviews in Number Theory as printed in Mathematical Reviews 1940–1972, Amer. Math. Soc. Providence, RI (1974). See N20-3, Vol. 4, 191–193.Google Scholar
14. [La1]
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig (1909), 2 volumes, reprinted by Chelsea Publishing Company, New York (1953).Google Scholar
15. [La2]
E. Landau, Über den Wienerschen neuen Weg zum Primzahlsatz, Sitzber. Preuss. Akad. Wiss., 1932, 514–521.Google Scholar
16. [Le1]
A.M. Legendre, Essai sur la théorie des nombres, 1. Aufl. Paris (Duprat) (1798).Google Scholar
17. [Le2]
A.M. Legendre, Essai sur la théorie des nombres, 2. Aufl. Paris (Courcier) (1808).Google Scholar
18. [S]
A. Selberg, An elementary proof of the prime-number theorem, Ann. of Math. (2) 50 (1949), 305–313; reprinted in Atle Selberg Collected Papers, Springer-Verlag, Berlin Heidelberg New York, 1989, vol 1, 379–387.
19. [Syl1]
J.J. Sylvester, On Tchebycheff’s theorem of the totality of prime numbers comprised within given limits, Amer. J. Math. 4 (1881), 230–247.
20. [Syl2]
J.J. Sylvester, On arithmetical series, Messenger of Math. (2) 21 (1892), 1–19 and 87–120.Google Scholar
21. [Tch1]
P.L. Tchebychef, Sur la fonction qui determine la totalité des nombres premiers inférieurs à une limite donnée, Mémoires préséntes à l’Académie Impériale des Sciences de St.-Pétersbourg par divers Savants et lus dans ses Assemblées, Bd. 6, S. (1851), 141–157.Google Scholar
22. [Tch2]
P.L. Tchebychef, Mémoire sur les nombres premiers, J. de Math. Pures Appl. (1) 17 (1852), 366–390; reprinted in Oeuvres 1 (1899), 49–70.Google Scholar
23. [VP]
C.J. de la Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers, Ann. Soc. Sci. Bruxelles 20 (1896), 183–256.Google Scholar
24. [W1]
N. Wiener, A new method in Tauberian theorems, J. Math. Physics M.I.T. 7 (1927–28), 161–184.
25. [W2]
N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), 1–100.